Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1995-1996. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 23)

CHUNG, P. v., Multiplicative functions satisfying the equation f(m2+n2) = (f(m))2 + (f(n))2

Multiplicative functions satisfying ... 31 for all positive integers q and k. So, the condition (C-3) is fulfilled. If p is prim and p = 1 (mod 4), then it is well known that there exist positive integers x and y such that k 2,2 p = x + y . By using (C), Lemma 4., we have f(p

k) = f(x

2 + y

2) = (f(x))

2 + (f{y))

2 = x

2+y

2 = p

k

, which verified the condition (C-2). So, we have completed the proof the necessity of the condition. Conversely, suppose that the conditions (C-l), (C-2) and (C-3) are satisfied for a multiplicative function /. It is well known that we can write m

2+n

2 = 2

kp"

1p%

3 • ••p

a

h

hql

ß

lql

ß

2 • • -q

2

a

ß', where pi and q

3 are primes, p

l = I (mod 4) and q

3 = 3 (mod 4) for i — 1, 2 . . ., h and j = 1, 2,..., s, and k > 0 and Q,-, ßi are positive integers. Then by the multiplicativity of / and the conditions (C-l), (C-2) and (C-3), /(m

2 + n

2) = f(2

k)f(

Pr )••• S

(PT)/(??"')''' Hi

2/'

) = 2"p? • • -vTll

0

1 •

' •?

2"' = m

2 + n

2 = (/(m))

2 + ()(n)f

. So, this completes the proof of the theorem. References [1] P. V. CHUNG, Multiplicative Functions / Satisfying the Equation /(m

2 +n

2) = f(m

2) + f(n

2). Mathematica Slovaca, Vol. 44 (1994), (to appear) [2] SPIRO, CLAUDIA A., Additive Uniwueness Sets for Arithmetic Functions, Journal of Number Theory, 42 (1992), 232—246.